Water is flowing through a horizontal pipe at a rate 0.2 m3/s. The pipe narrows from a radius of 10 cm to 4 cm. What is the change in pressure?
To calculate the change in pressure, we can use the Bernoulli's equation, which states that the total pressure of a fluid remains constant along a streamline.
The Bernoulli's equation is given by:
P₁ + 1/2ρv₁² + ρgh₁ = P₂ + 1/2ρv₂² + ρgh₂
Where:
P₁ and P₂ are the initial and final pressures respectively,
ρ is the density of the fluid,
v₁ and v₂ are the initial and final velocities respectively,
g is the acceleration due to gravity, and
h₁ and h₂ are the initial and final heights respectively.
Since the water is flowing horizontally, we can ignore the height terms:
P₁ + 1/2ρv₁² = P₂ + 1/2ρv₂²
Let's calculate the velocities at both ends of the pipe:
The initial velocity v₁ can be calculated using the equation for the volume flow rate:
Q = Av
0.2 m³/s = π(0.1 m)²v₁
v₁ = 0.2 m³/s / (π(0.1 m)²) ≈ 6.366 m/s
Similarly, the final velocity v₂ can be calculated:
Q = Av
0.2 m³/s = π(0.04 m)²v₂
v₂ = 0.2 m³/s / (π(0.04 m)²) ≈ 79.58 m/s
Now we can substitute the values into the Bernoulli's equation:
P₁ + 1/2ρ(6.366 m/s)² = P₂ + 1/2ρ(79.58 m/s)²
We know that the density of water is approximately 1000 kg/m³, so we can substitute this value:
P₁ + 1/2(1000 kg/m³)(6.366 m/s)² = P₂ + 1/2(1000 kg/m³)(79.58 m/s)²
Simplifying the equation and rearranging to solve for the change in pressure (P₂ - P₁), we get:
P₂ - P₁ = 1/2(1000 kg/m³)(79.58 m/s)² - 1/2(1000 kg/m³)(6.366 m/s)²
Calculating the values on the right side of the equation gives us the change in pressure.